Le Bruyn, Lieven; Reichstein, Zinovy Smoothness in algebraic geography. (English) Zbl 1032.16012 Proc. Lond. Math. Soc., III. Ser. 79, No. 1, 158-190 (1999). The most naive way to understand a finite-dimensional associative algebra is to find a basis and analyze its multiplication table. In the modern incarnation, one considers the scheme of all associative \(n\)-dimensional algebras over the field \(k\) as a subscheme of affine space \(\operatorname{Hom}_k(k^n\otimes k^n,k^n)\). Then \(\text{GL}_n(k)\) acts on the \(k\)-rational points of the scheme so that orbits can be interpreted as isomorphism classes of \(n\)-dimensional algebras. So the isomorphism classes of algebras can be understood as a subvariety of \(\operatorname{Hom}_k(k^n\otimes k^n,k^n)\), which is denoted by \(\text{Alg}_r\). The paper analyses the smoothness of some orbits. The particular case where \(X_n\) is the component of the matrix ring is of importance. The paper answers negatively a question of Seshadri and shows that \(X_n\) is singular for \(n\geq 3\). The name Algebraic Geography was given by Flanigan which referred to the study of the variety \(\text{Alg}_r\) in this fashion. Reviewer: Eduardo Marcos (São Paulo) Cited in 2 Documents MSC: 16G10 Representations of associative Artinian rings 16R30 Trace rings and invariant theory (associative rings and algebras) 16P10 Finite rings and finite-dimensional associative algebras 16S80 Deformations of associative rings Keywords:algebraic geography; smooth orbits; deformations; finite-dimensional algebras; schemes PDF BibTeX XML Cite \textit{L. Le Bruyn} and \textit{Z. Reichstein}, Proc. Lond. Math. Soc. (3) 79, No. 1, 158--190 (1999; Zbl 1032.16012) Full Text: DOI